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Find the greatest values of sin (x − π /3 ) sin ( π/6 + x) and also find the corresponding value of x for which it is greatest, where x ∈ [π, 3π /2 ]. |
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Answer» GIVEN : SIN (x − π /3 ) sin ( π/6 + x) To Find : greatest values corresponding VALUE of x for which it is greatest, where x ∈ [π/2, 3π /2 ]. Solution: f(x) = sin (x − π /3 ) sin ( π/6 + x) f'(x) = sin (x − π /3) Cos ( π/6 + x) + cos (x − π /3 ) sin ( π/6 + x) = Sin(x - π /3 + π/6 + x ) = Sin( 2X - π/6) f'(x) = 0 Sin( 2x - π/6) = 0 => x = π/12 , 7π/12 , 13π/12 , 19π/12 7π/12 , 13π/12 ∈ [π/2, 3π /2 ] f''(x) = Cos ( 2x - π/6) x = 7π/12 f''(x) is -ve Hence max value at x = 7π/12 f(7π/12) = sin (7π/12 − π /3 ) sin ( π/6 + 7π/12) = sin( 3π/12) sin (9π/12) = sin( π/4) sin (3π/4) = (1/√2) (1/√2) = 1/2 Max value = 1/2 at x = 7π/12 if Question is x ∈ ∈ [π , 3π /2 ] then at 3π /2 is max value sin (3π/2 − π /3 ) sin ( π/6 + 3π/2) = sin( 7π/6)sin(5π/3) = √3/4 Learn More: find the local maxima or minima of the function: f(x)= sinx + cosx ... examine the maxima and minima of the function f(x)=2x³-21x²+36x ... |
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